Broken Lorentz symmetry and violation of the Wiedemann-Franz law in topological insulators

We study a two-dimensional topological insulator in the presence of a static nonuniform gravitational field, which mimics the variations in the temperature distribution. We derive an effective boundary free energy functional for the gravitational field and show that, in contrast to the case of massive Dirac fermions, the addition of a Newtonian mass term significantly modifies the quantum anomalous behavior of the system. A nonzero bulk thermal current appears, which violates the Wiedemann-Franz law. The systematic approach we develop to calculate the contribution of edge states to thermal Hall conductivity and energy magnetization can easily be extended to other models.